/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 101 $$ \text { How do the graphs o... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 101

$$ \text { How do the graphs of } f(x) \text { and } f(x+10) \text { differ? } $$

Short Answer

Expert verified
The graph of \( f(x+10) \) is shifted 10 units left compared to \( f(x) \).

Step by step solution

01

Identify the Functions

The functions given are \( f(x) \) and \( f(x+10) \). They represent a standard function and its horizontal shift, respectively.
02

Understand Horizontal Shifts

When you have a function \( f(x) \) and you add a constant inside the argument, like in \( f(x+c) \), the graph will shift horizontally. The rule is that \( f(x+c) \) will move the graph \( c \) units to the left if \( c > 0 \).
03

Apply the Horizontal Shift to f(x+10)

The function \( f(x+10) \) is \( f(x) \) shifted by 10 units to the left. Adding 10 inside the function affects only the x-coordinate by moving each point on the graph of \( f(x) \) 10 units to the left.
04

Conclusion by Comparing Graphs

The graph of \( f(x+10) \) is identical in shape and size to \( f(x) \), but every point on the graph of \( f(x) \) has been moved 10 units to the left in the graph of \( f(x+10) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Horizontal Shifts
One of the basic transformations of functions is the horizontal shift. This transformation involves moving the graph of a function left or right along the x-axis, without changing its shape. When you see a function written as \( f(x + c) \), it indicates a horizontal shift. Here, \( c \) is the number of units the graph will shift.
  • If \( c \) is positive, the graph shifts to the left by \( c \) units.
  • If \( c \) is negative, the graph shifts to the right by \(-c\) units.
For example, with the function \( f(x+10) \), the entire graph of \( f(x) \) moves 10 units to the left. Although the position changes, the overall shape and size of the graph remain the same.
Graph Comparison
When comparing graphs of functions, look beyond the shapes and examine the transformations applied. The primary transformation in our exercise is the horizontal shift. Despite the horizontal movement, the graphs of \( f(x) \) and \( f(x+10) \) are identical in shape.
This means:
  • Both graphs have the same height and width at corresponding points.
  • Each point on \( f(x+10) \) is exactly 10 units left of its corresponding point on \( f(x) \).
Understanding graph comparison helps in visualizing the effect of transformations and in predicting how changes to functions will affect their graphs.
Function Notation
Function notation is a convenient way to express mathematical relationships and transformations. By using a function notation like \( f(x) \), we establish a clear rule that explains how to arrive at outputs from the given inputs, x-values.
Here are some key points about function notation:
  • \( f(x) \) represents the original function, where each x-value is plugged into the function rule to obtain a corresponding \( y \)-value.
  • Functions can be transformed by altering the inputs, as seen with \( f(x+c) \), resulting in predictable shifts in the graph.
  • Clear function notation makes it easier to understand and predict the effect of adjustments, such as shifts or stretches, on the graph.
Using function notation not only clarifies transformations applied to the function but also simplifies communication and understanding in mathematics.

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Most popular questions from this chapter

GENERAL: Stopping Distance A car traveling at speed \(v\) miles per hour on a dry road should be able to come to a full stop in a distance of $$ D(v)=0.055 v^{2}+1.1 v \text { feet } $$ Find the stopping distance required for a car traveling at: \(60 \mathrm{mph}\).

Can the graph of a function have more than one \(x\) intercept? Can it have more than one \(y\) -intercept?

Find the \(x\) -intercept \((a, 0)\) where the line \(y=m x+b\) crosses the \(x\) -axis. Under what condition on \(m\) will a single \(x\) -intercept exist?

SOCIAL SCIENCE: Immigration The percentage of immigrants in the United States has changed since World War I as shown in the following table. \begin{tabular}{llllllllll} \hline Decade & 1930 & 1940 & 1950 & 1960 & 1970 & 1980 & 1990 & 2000 & 2010 \\\ \hline Immigrant Percentage & \(11.7\) & \(8.9\) & 7 & \(5.6\) & \(4.9\) & \(6.2\) & \(7.9\) & \(11.3\) & \(12.5\) \\ \hline \end{tabular} a. Number the data columns with \(x\) -values \(0-8\) (so that \(x\) stands for decades since 1930 ) and use 0 quadratic regression to fit a parabola to the data. State the regression function. [Hint: See Example 10.] b. Use your curve to estimate the percentage in \(2016 .\)

The following function expresses dog-years as 15 dog-years per human-year for the first year, 9 dog-years per human-year for the second year, and then 4 dog-years per human-year for each year thereafter. \(f(x)=\left\\{\begin{array}{ll}15 x & \text { if } 0 \leq x \leq 1 \\\ 15+9(x-1) & \text { if } 12\end{array}\right.\)
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